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Homework Help: Force of magnetic attraction between hemispheres

  1. Nov 9, 2017 #1
    1. The problem statement, all variables and given/known data

    Calculate the force of magnetic attraction between the northern and southern hemispheres of a uniformly charged spinning spherical shell, with radius R, angular velocity ω and surface charge density σ.

    2. Relevant equations
    Maxwell Tensor : Tm =
    (1/μ) * ((B*n)B - (1/2)*B2 n)

    B_in =
    B_out =
    μm/(4πr3) * (2cosθ r + sinθ θ)
    m = (4/3)*πR3(σωR)

    3. The attempt at a solution
    I do see in my solution that :
    1-We use a surface consisting of the entire equatorial plane, CLOSING IT with a hemispherical surface at infinity where (since the field is zero out there) contribution is zero.
    for r>R :
    B = μm/(4πr3) θ = - μm/(4πr3) z (Why ?! )
    We have a equatorial circular disk. We use B inside and da = rdrdφ. Direction is in z-axis!
    I don't want the solution because I already have it but MY QUESTIONS are :

    A) How should think and imagine this spheres ?!
    Should I think like this and use those formulas :

    B) Why we do write B = μm/(4πr3) θ = - μm/(4πr3) z
    Why is θ =-z in this case ?! I dont get it !

    I know the rest. That is just integrating
  2. jcsd
  3. Nov 9, 2017 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    I'm not quite following what you are asking here. Can you reword the question?

    It is because you are using spherical coordinates with θ measured from the positive z axis. The unit vector θ points in the direction of increasing θ. For a point on the equatorial plane of the sphere, what is the direction of θ?
  4. Nov 10, 2017 #3
    I've got it !

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